Label the following statements as true or false. (a) Every quadratic form is a bilinear form. (b) If two matrices are congruent, they have the same eigenvalues. (c) Symmetric bilinear forms have symmetric matrix representations. (d) Any symmetric matrix is congruent to a diagonal matrix. (e) The sum of two symmetric bilinear forms is a symmetric bilinear form. (f) Two symmetric matrices with the same characteristic polynomial are matrix representations of the same bilinear form. (g) There exists a bilinear form H such that H(x, y)≠0 for all x and y. (h) If V is a vector space of dimension n, then dim(B(V )) = 2n. (i) Let H be a bilinear form on a finite-dimensional vector space V with dim(V) >1. For any x ∈V, there exists y ∈V such that y ≠0 , but H(x, y) = 0. (j) If H is any bilinear form on a finite-dimensional real inner product space V, then there exists an ordered basis β for V such that ψβ(H) is a diagonal matrix.
Label the following statements as true or false.
(a) Every quadratic form is a bilinear form.
(b) If two matrices are congruent, they have the same eigenvalues.
(c) Symmetric bilinear forms have symmetric matrix representations.
(d) Any symmetric matrix is congruent to a diagonal matrix.
(e) The sum of two symmetric bilinear forms is a symmetric bilinear form.
(f) Two
(g) There exists a bilinear form H such that H(x, y)≠0 for all x and y.
(h) If V is a
(i) Let H be a bilinear form on a finite-dimensional vector space V with dim(V) >1. For any x ∈V, there exists y ∈V such that y ≠0 , but H(x, y) = 0.
(j) If H is any bilinear form on a finite-dimensional real inner product space V, then there exists an ordered basis β for V such that ψβ(H) is a diagonal matrix.
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